A, B, C ~ unif(0,1), find prob of B<2sqrtAC

We are tasked with finding the probability that \( B < 2\sqrt{AC} \), where \( A \), \( B \), and \( C \) are independent random variables uniformly distributed on \([0, 1]\), i.e., \( A, B, C \sim \text{Unif}(0,1) \).

### Step 1: Setting up the problem

We need to compute the probability:
\[
P(B < 2\sqrt{AC})
\]
Since \( A \), \( B \), and \( C \) are uniformly distributed on \([0, 1]\), their joint distribution is given by:
\[
f_{A,B,C}(a,b,c) = 1 \quad \text{for} \quad 0 \leq a,b,c \leq 1
\]
The probability we are interested in can be expressed as a triple integral over the region where \( B < 2\sqrt{AC} \):
\[
P(B < 2\sqrt{AC}) = \int_0^1 \int_0^1 \int_0^{2\sqrt{ac}} 1 \, db \, da \, dc
\]

### Step 2: Solving the integral

We will solve this integral step by step.

1. **Integrating with respect to \( b \):**
\[
\int_0^{2\sqrt{ac}} 1 \, db = 2\sqrt{ac}
\]

2. **Integrating with respect to \( a \) and \( c \):**
Now, we need to compute the double integral:
\[
P(B < 2\sqrt{AC}) = \int_0^1 \int_0^1 2\sqrt{ac} \, da \, dc
\]
We can factor the integrals because they are independent:
\[
P(B < 2\sqrt{AC}) = 2 \left( \int_0^1 \sqrt{a} \, da \right) \left( \int_0^1 \sqrt{c} \, dc \right)
\]

3. **Evaluating the integrals:**
The integral of \( \sqrt{x} \) over \([0, 1]\) is:
\[
\int_0^1 \sqrt{x} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_0^1 = \frac{2}{3}
\]

Thus, we get:
\[
P(B < 2\sqrt{AC}) = 2 \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{9}
\]

### Final Answer:
\[
P(B < 2\sqrt{AC}) = \frac{8}{9}
\]

Would you like further details or clarifications on any part of the solution?

### Related Questions:
1. How do you calculate the probability for a more complex region in three-dimensional space?
2. What are the properties of the uniform distribution used in these types of probability problems?
3. How does the assumption of independence affect the setup of the triple integral?
4. Can the method used here be applied to non-uniform distributions?
5. How does this approach generalize to other inequalities involving random variables?

### Tip:
Always verify that the region of integration aligns with the conditions set by the inequality before performing the integral.

A, B, C ~ unif(0,1), find prob of B < 2sqrtAC

Find the probability that B < 2√(AC) where A, B, and C are independent and uniformly distributed on [0, 1]. This problem involves solving a triple integral and applies probability theory, uniform distribution, and multivariable calculus. Suitable for undergraduate students studying probability theory.

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